Energy of Fuzzy Labeling Graph [EFl(G)]- Part I · Energy of Fuzzy Labeling Graph [EF l (G)]- Part I. S.Vimala and A.Nagarani. Abstract—In this paper, we introduced energy of fuzzy

International Journal of Scientific & Engineering Research, Volume 7, Issue 4, April-2016 1910 ISSN 2229-5518

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Energy of Fuzzy Labeling Graph [EFl(G)]- Part I S.Vimala and A.Nagarani

Abstract—In this paper, we introduced energy of fuzzy labeling graph and its denoted by [EFl(G)]. We extend the concept of fuzzy labeling graph to the

energy of fuzzy labeling graph [EFl(G)]. This result tried for some fuzzy labeling graphs such as butterfly graph, book graph, wheel graph, caterpillar

graph, theta graph, Hamiltonian circuit graph , 𝐾2 𝐽 𝐾2�� graph, 𝐾3 𝐽 𝐾3 �� graph and studied the characters.

Keyword: Labeling, fuzzy labeling graph, energy graph, energy of fuzzy labeling graph.

—————————— ——————————

1 INTRODUCTION

Most graph labeling methods trace their origin to

one introduced by Rosa [10] in 1967,and one given by Graham

and Sloane [5] in 1980.Pradhan and Kumar [10] proved that

graphs obtained by adding a pendent vertex of hair cycle 𝐶𝑛

ʘ𝐾1are graceful if n≡ 0( mod4m ).They further provide a rule

for determining the missing numbers in the graceful labeling

of 𝐶𝑛ʘ𝐾1and of the graph obtained by adding pendent edges

to each pendent vertex of 𝐶𝑛 ʘ𝐾1. Abhyanker[2] proved that

the graph obtained by deleting the branch of length 1 from an

olive tree with 2n branches and identifying the root of the edge

deleted tree with a vertex of a cycle of the form 𝐶2𝑛+3 is

graceful.In 1985 Koh and Yap[7] generalized this by defining a

cycle with a 𝑃𝑘 -chord to be a cycle with the path 𝑃𝑘 joining

two nonconsecutive vertices of the cycle.

Fuzzy relation on a set was defined by Zadeh in 1965.

Based on fuzzy relation the first definition of a fuzzy graph

was introduced by Rosenfeld and Kaufmann in 1973. Fuzzy

graph have many more applications in modeling real time

system where the level of information inherent in the system

varies with different levels of precision.

The concept of energy graph was defined by I. Gutman in

1978. The energy, E(G), of a graph G is defined to be the sum of

the absolute values of its eigen values. Hence if A(G) is the

adjacency matrix of G, and λ1, λ2, ….., λ n are the eigen values

of A(G), then E(G) = ∑ |𝜆𝑖𝑛𝑖=1 |. The set {λ1,….., λn } is the

spectrum of G and denoted by Spec G. The upper and lower

bound for energy was introduced by R. Balakrishnan[3]. The

totally disconnected graph 𝐾𝑛𝑐 has zero energy while complete

graph 𝐾𝑛 with the maximum possible number of edges has

energy 2(n-1). It was therefore conjectured in P.Prabhan and A.

Kumar [9] that all graphs have energy at most 2(n-1). But then

this was disproved in A. Nagoorgani, and D. Rajalaxmi (a)

Subahashini, [8].

We generalise the energy of fuzzy labeling graph EFl(G)

for butterfly graph (13, 21) book graph (8,10), wheel graph(6,

10), (7,12), caterpillar graph(11,10) ,theta graph(6,7),

Hamiltonian circuit graph(12 17), 𝐾2 𝐽 𝐾2�� (4,5) graphs, 𝐾3 𝐽 𝐾3 ��

(6,12)graph. And also find upper bounds and lower bounds for

energy of fuzzy labeling graphs.

2 PRELIMINARIES

2.1 Labeling A labeling of a graph is an assignment of values to the vertices

and edges of a graph.

2.2 Vertex Labeling

Given a graph G, an injective function f :V(G)→ N has been

called a vertex labeling of G.

2.3 Edge Labeling

———————————————— • S.Vimala,Assistant Professor, Mother Teresa Women’s University,India.

E-mail: [email protected] • A.Nagarani Mother Teresas Women’s University, India. E-mail:

[email protected]

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An edge labeling of a graph is a bijection from E(G) to the set

{1, 2. … ,|E(G| }.

2.4 Fuzzy Graph

A fuzzy graph G =( 𝜎,𝜇 ) is a pair of function 𝜎: V →[0,1] and 𝜇

: V×V → [0,1], where for all u,v ∈ V,we have 𝜇 ( u,v ) ≤ 𝜎 (u) ⋀

𝜎 (v).

2.5 Fuzzy Labeling Graph

A graph G = (𝜎,𝜇 ) is said to be a fuzzy labeling graph if 𝜎 :V

→[0,1] and u : V×V [0,1] is bijective such that the membership

value of edges and vertices are district and u ( u,v) < 𝜎 (u) ⋀

𝜎(V) for all u,v ∈ V , [7], [6]

2.6 Energy Graph

Energy of a simple graph G = (V,E) with adjacency matrix A is

defined as the sum of absolute values of eigenvalues of A. It

is denoted by E (G). E(G) =∑ |𝜆𝑖|𝑛𝑖=1 where λ i is an eigenvalues

of A, i =1, 2, . . . , n.

2.7 Energy of Fuzzy Graph

Let G= (V,𝜎,𝜇) be a fuzzy graph and A be its adjacency matrix.

The eigenvalues of A are called eigenvalues of G. The

spectrum of A is called the spectrum of G. It is denoted by Spec

G. Let G = (V, 𝜎,𝜇 ) be a fuzzy graph and A be its adjacency

matrix. Energy of G is defined as the sum of absolute values of

eigenvalues of G[1].

3 MAIN RESULTS 3.1 Energy of Fuzzy Labeling Graph

The following three conditions are true if the graph is called a

energy of fuzzy labeling graph. We denote EFl(G) be a energy

of fuzzy labeling graph.

(i) EFl(G) =∑ |𝜆𝑖|𝑛𝑖=1

(ii) μ (u, v ) > 0

(iii) μ (u, v ) < σ(u) Ʌ σ(v)

(iv) Let Fl (G) be a fuzzy labeling graph with |V| = n

vertices and μ = {e1,…..em }. If mi = μ(ei) ,

then �2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >

� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G) .

Theorem 1. Let Fl (G) be a fuzzy labeling graph with |V| = n

vertices and μ = {e1,…..em }.If mi = μ(ei) , then

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)

Proof. Upper bound for energy of fuzzy graph is same as an

energy of fuzzy labeling graph. [1]

ie ., EFl(G) � 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1

Lower bound

[EFl(G) ]2 = Fl (∑ |𝜆𝑖|𝑛𝐼=1 )P

2

=Fl ∑ |𝜆𝑖|𝑛𝑖=1 P

2 +2 ∑ |𝜆𝑖𝜆𝑗|1≤𝑖<𝑗≤𝑛

= Fl 2 ∑ 𝑚𝑖2𝑚

𝑖=1 + 2 𝑛 (𝑛−1)2

AM {|𝜆𝑖𝜆𝑗|}

AM {|𝜆𝑖𝜆𝑗|} GM {|𝜆𝑖𝜆𝑗|} if Fl for 1≤ 𝑖 < 𝑗 ≤ 𝑛

EFl(G) ≤ � 2∑ 𝑚𝑖2𝑚

𝑖=1 + 𝑛(𝑛 − 1)GM {|𝜆𝑖𝜆𝑗|}

GM {|𝜆𝑖𝜆𝑗|} = (∏ �𝜆𝑖𝜆𝑗�1≤𝑖<𝑗≤𝑛 ) 2/n(n-1)

=( ∏ |𝜆𝑖𝜆𝑗|𝑛𝑖=1 P

n-1 )2/n(n-1)

= (∏ |𝜆𝑖|𝑛𝑖=1 )2/n = |A | 2/n

EFl(G) �2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1

Therefore, �2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1

>� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)

Here, we found energy of fuzzy labeling graphs EFl(G) ,

upper bound, and lower bounds of some graphs like

Hamiltonian Circuit graph, Book Graph, Caterpillar

Graph, Wheel Graph, Theta Graphand etc., and also

made between them.

Theorem 2. Fuzzy Labeling of 𝑲𝟐J𝑲𝟐− satisfy

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)

for 4 vertices and 5 edges.

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Adjacency matrix of the fuzzy labeling graph of (above) is

A = �0 0.003

0.003 00.002 00.006 0.007

0.002 0.0060 0.007

0 0.0100.010 0

�

The eigen values are = -0.0104, -0.0058, 0.003, 0.0159

The energy of the graph EFl(G) is = 0.0324

The upper bound of the graph is EFl(G) = 0.03979

The lower bound of the graph is EFl(G) = 9.6439

Theorem 3. Fuzzy Labeling of Hamiltonian Circuit graph statisfy

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)


The eigen values are = -0.2721,0.2721,-

0.0880,0.0900,0.0696,0.0361,0.0211,0,0,-0.0360,

-0.0234,-0.0194



The lower bound of the graph is EFl(G) =19.74273

Theorem 4. Fuzzy Labeling of Book Graph statisfy

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)


The eigen values are = -0.0018,-0.0008,-0.0003,-

0.0001,0.0001,0.0003,0.0008,0.0018




Theorem 5. Fuzzy Labeling of Caterpillar Graph satisfy

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)


Solution of Caterpillar graph generalized. Below values

verified for the same.

The eigen values are = 0.0374,-0.0374,0,0,-0.1827,0.1827,-

0.0918,0.0918,0.0372, -0.0372,0




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Theorem 6. Fuzzy Labeling of Wheel Graph satisfy EFl(G) ˂

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)

vertices and 10 edges.

The eigen values are = -0.0218,-0.0158,-

0.0069,0.0008,0.0063,0.0374


The upper bound of the graph is EFl(G) =0.5396


Theorem 7. Fuzzy Labeling of Wheel Graph satisfy EFl(G) ˂

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)

vertices and 12 edges.

The eigen values are =-0.0286,-0.0208,-0.0089,-

0.0021,0.0029,0.0094,0.0481




Theorem 8. Fuzzy Labeling of 𝑲𝟑J𝑲𝟑− satisfy

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)


The eigen values are = -0.0523,-0.0177,-

0.0079,0.0005,0.0148,0.0625


The upper bound of the graph is EFl(G)= 0.5396

The lower bound of the graph is EFl(G)= 11.3149

Theorem 9. Fuzzy Labeling of Theta Graph satisfy

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)

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EFl(G) for 5 vertices and 7 edges.

The eigen values are = 0.2532,-0.2493,-0.0658,0.0515,0,0



The lower bound of tha graph is EFl(G) =11.32

Theorem 10. Fuzzy Labeling of Butterfly Graph satisfy

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G)


The eigen values are = -0.0793,-0.0612,-0.0373,-0.0319,

-0.0179,0,0,0,0.0028,0.0190,0.0374, 0.0402,0.0183


The upper bound of the graph is EFl(G)= 0.41870


4 CONCLUSION

In this paper, we find some energy of fuzzy labeling graphs

and to the energy level is

�2∑ 𝑚𝑖2 + 𝑛(𝑛 − 1) |𝐴|

2𝑛 𝑚

𝑖=1 >� 2( ∑ 𝑚𝑖2 )𝑛 𝑚

𝑖=1 > EFl(G). Our

future work are s to apply п electron in fuzzy labeling and also

discuss with other labeling graphs .

REFERENCES [1] Anjali Narayanan,Sunil Mathew , Energy of a fuzzy graphs,

Annals of fuzzy mathematics and Informatics ,vol 6, no 3. pp.

455-465.

[2] V.J Abhyankar, Direct methods of gracefully labeling

graphs, Ph.D thesis, University of Mumbai, 2002.

[3] R. Balakrishnan, The energy of a graph, Linear Algebra

Appl. 387 (2004),287-295.

[4] R.L. Graham and N. J. A. Sloane, On additives bases and

harmonious graphs, Alg .Discrete Methods, 1 (1980) 382-404.

[5] K.A Germina and Hameed, Thomas Zaslavsky, On

products and line graphs of signed graphs, their eigenvalues

and energy, Linear Algebra Appl. 435 (2011) 2432-2450.

[6] R .Jebesty Shajila and S.Vimala, Some remarks of fuzzy

graceful labeling on special graphs, International Organization

Of Scientific research ,accepted and communicates in March

issue.

[7]K.M Koh and K.Y.Yap. Graceful numberings of cycles with

a chord , Bull . Inst. Math. Acad Sinica, 12(1985) 41-48

[8] A. Nagoorgani, and D. Rajalaxmi (a) Subahashini,

Properties of fuzzy labeling graph, Applied Mathematical

Science, 6, no. 70 (2012),3461-3466.

[9] P.Prabhan and A. Kumar, Graceful hairy cycles with

pendent edges and some properties of cycle and cycle related

graphs,Bull Calcutta Math. Soc 103 (2011), no. 3, 233-246.

[10] A.Rosa On certain valuations of the vertices of a graph,

Theory of graphs ,Gordon and Breach, N. Y. and Dunod Paris

(1967) 349-355.

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Energy of Fuzzy Labeling Graph [EFl(G)]- Part I · Energy of Fuzzy Labeling Graph [EF l (G)]- Part I. S.Vimala and A.Nagarani. Abstract—In this paper, we introduced energy of fuzzy